Transcript Process modeling - Monmouth University

SE-561
Formal Methods in Software
Petri Nets - I
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Petri nets
• The classical Petri net was invented by Carl Adam Petri in
1962.
• A lot of research has been conducted (>10,000 publications).
• Until 1985 it was mainly used by theoreticians.
• Since the 80-ties the practical use is increasing because of the
introduction of high-level Petri nets and the availability of
many tools.
• High-level Petri nets are Petri nets extended with
– color (for the modeling of attributes)
– time (for performance analysis)
– hierarchy (for the structuring of models)
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The classical Petri net model
A Petri net is a network composed of places ( ) and transitions ( ).
t2
t1
p2
p1
t3
p4
p3
• Connections, called arcs, are directed and between a place and a
transition.
• Tokens ( ) are the dynamic objects.
• The state of a Petri net, called marking, is determined by the
distribution of tokens over the places.
• Initial marking (1, 2, 0, 0)
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p1
p4
t1
p2
p3
Transition t1 has three input places (p1, p2 and p3) and two
output places (p3 and p4).
Place p3 is both an input and an output place of t1.
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Enabling condition
• Transitions are the active components, while places and tokens are
passive.
• A transition is enabled if each of the input places contains tokens.
t1
t2
• Transition t1 is not enabled, transition t2 is enabled.
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Firing
An enabled transition may fire.
Firing corresponds to consuming tokens from the input places and
producing tokens for the output places.
t2
t2
Firing is atomic.
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Example
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Petri Net Structures
• sequence of events/actions:
t1
t2
t3
t2
t3
t4
t5
• concurrent execution:
t1
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Non-determinism
t1
Two transitions fight for the same token: conflict.
Even if there are two tokens, there is still a conflict.
t2
Synchronization
t1
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Modeling
• States of a process or a system are modeled by tokens in places and
state transitions leading from one state to another are modeled by
transitions.
• Tokens represent objects (humans, goods, machines), information,
conditions or states of objects.
• Places represent buffers, channels, geographical locations,
conditions or states.
• Transitions represent events, transformations or transportations.
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Example: Traffic light
red
yr
yellow
rg
gy
green
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Two traffic lights
red1
red2
yr1
yr2
yellow1
rg1
gy1
yellow2
rg2
gy2
green1
green2
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Two safe traffic lights
red1
red2
safe
yr1
yr2
yellow1
rg1
gy1
yellow2
rg2
gy2
green1
green2
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Two safe and fair traffic lights
red1
red2
safe2
yr1
yr2
yellow1
rg1
yellow2
gy1
rg2
gy2
safe1
green1
green2
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Reachability
• Reachable marking
A marking is said to be reachable if it can be reached by firing a
sequence of enabled transitions form the initial marking of a Petri
net.
• Petri net token game
http://psim.tm.tue.nl/staff/wvdaalst/Downloads/pn_applet/pn_applet.html
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Exercise: readers and writers
begin
receive_mail
mail_box
rest
rest
type_mail
read_mail
send_mail
ready
• How many states are reachable?
• How to model the situation with 2 writers and 3 readers?
• How to model a "bounded mailbox" (buffer size =4)?
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Restaurant Scenario
Waiter
free
Customer 1
Customer 2
Take
order
Take
order
wait
Order
taken
wait
eating
eating
Serve food
Tell
kitchen
Serve food
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A Puzzle: Crossing River
 A ferry-man has to bring a goat, a cabbage, and a wolf safely from
the left bank to the right bank of a river.
 The ferry-man can cross the river alone or with exactly one of these
three passengers.
 At any time, either the ferry-man should be on the same bank as the
goat, or the goat should be alone on a bank.
 Otherwise, the goat could go ahead and eat the cabbage or the
wolf may eat the goat
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Modeling
• We are going to model the situation with a Petri net.
• The puzzle mentions the following objects:
– Man, wolf, goat, cabbage, boat.
• The puzzle mentions the following actions:
– Crossing the river, wolf eats goat, goat eats cabbage.
• Objects and their states are modeled by places.
• Actions are modeled by transitions.
• Actually, we can omit the boat, because it is always going to be on
the same side as the man.
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Places
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Crossing the river (left to right)
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Crossing the river (left to right)
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Crossing the river (left to right)
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Crossing the river (right to left)
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Wolf eats goat
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Formal Definition of a Petri Net
A Petri net N is a tuple N = {P, T, I, O, M0}, where
 P is a finite set of places,
 T is a finite set of transitions,
 Places P and transitions T are disjoint (P ∩ T = f),
 I: P × T  N (N = {0, 1, 2, …})is the pre-incidence function representing input
arcs,
 O: T × P  N (N = {0, 1, 2, …})is the post-incidence function representing
output arcs,
 M0 : P  N is the initial marking representing the initial distribution of tokens.
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Example
t1
p1
p2
t2
•
•
•
•
•
t3
P = {p1, p2}
T = {t1, t2, t3}
I(t1) = (1, 1), I(t2) = (2, 0), I(t3) = (0, 2)
O(t1) = ?
M0 = (3, 2)
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Transition firing
• A transition t is enabled at marking Mi if and only if
Mi ≥ I(t)
• Suppose that the firing of t takes the Petri net from Mi to Mi. Then
Mj = Mi - I(t) + O(t)
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t1
p1
p2
t2
•
•
•
•
•
t3
P = {p1, p2}
T = {t1, t2, t3}
I(t1) = (1, 1), I(t2) = (2, 0), I(t3) = (0, 2)
O(t1) = (1, 0), O(t2) = (0, 1), O(t3) = (0, 1)
M0 = (3, 2)
Mj = Mi - I(t) + O(t)
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